A platform that rolls on wheels and has inertia mp is attached to a wall by a horizontal spring of spring constant k. A load of inertia ml sits on the platform, and the coefficient of static friction between platform and load is μs. If you pull the platform away from the wall so that the spring is stretched a distance x from its relaxed length and then let go, the platform-spring combination bounces back and forth.

Question

If you want the load to stay in place held only by friction, what is the maximum distance xmax you can stretch the spring from its relaxed length and have the load stay on when you release the platform-spring combination?

Answer 1

max spring force = -k x
a = -k x/(mp+ml)

normal force between = ml g
max friction force = us ml g
that must result in acceleration a
a = us ml g/ml = -k x/(mp+ml)
so
us g =-k x/(mp+ml)